All of the 5th grade teachers and students from Covington went on a field trip to an archaeology museum. Tickets were $$7.50$ each for teachers and $$2.00$ each for students, and the group paid $$29.00$ in total. A few weeks later, the same group visited a science museum where the tickets cost $$30.00$ each for teachers and $$8.50$ each for students, and the group paid $$119.50$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7.5x+2y = 29}$ ${30x+8.5y = 119.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-30x-8y = -116}$ ${30x+8.5y = 119.5}$ Add the top and bottom equations together. $ 0.5y = 3.5 $ $ y = \dfrac{3.5}{0.5}$ ${y = 7}$ Now that you know ${y = 7}$ , plug it back into $ {7.5x+2y = 29}$ to find $x$ ${7.5x + 2}{(7)}{= 29}$ $7.5x+14 = 29$ $7.5x = 15$ $x = \dfrac{15}{7.5}$ ${x = 2}$ You can also plug ${y = 7}$ into $ {30x+8.5y = 119.5}$ and get the same answer for $x$ ${30x + 8.5}{(7)}{= 119.5}$ ${x = 2}$ There were $2$ teachers and $7$ students on the field trips.